This has been discussed to death, but I'll summarize what I've learned. Anonymous individuals don't get to vote. People who vote without an ID claim a name. At the polling station, they (the poll workers) look for that name on the voter roll. In many states, the voter is required to know an additional piece of information, like their mailing address on the voter roll. If you're not on the roll, then you don't get to vote -- or in some places, you get a provisional registration.

It's possible to exploit this: you can claim somebody else's name, and apparently get away with casting their vote. If that person also votes, then that's easily detected. If the jurisdiction doesn't have a way to separate provisionally-authentic ballots from ID-verified ballots, then the election only needs to be re-done if the number of fraudulent votes is large enough to impact the outcome of the election. Elsewhere, both provisional ballots would be spoiled.